Singularities in Geometric Variational Problems

几何变分问题中的奇点

基本信息

批准号：
1951070
负责人：
Luca Spolaor
金额：
$ 14.99万
依托单位：
University of California-San Diego
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2019
资助国家：
美国
起止时间：
2019-07-01 至 2022-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1951070&HistoricalAwards=false
关键词：
Singularities Geometric Variational Problems

项目摘要

The study of geometric variational problems -- the `problem' here means finding an object that minimizes some geometrically-defined notion of energy -- is one of the oldest and most fascinating topics in mathematics, dating back at least to the first "Nobel prize for mathematics" (the Fields Medal) awarded to J. Douglas in 1932 for his advances in this field. Solutions of geometric variational problems can describe equilibrium configurations of physical systems or socio-economical models -- situations where we expect real-world systems to naturally reach a minimum-energy configuration. As well, in the mathematical field of topology -- where objects are regarded as equivalent no matter how they are bent or stretched -- the solutions to variational problems can provide preferred representatives within such large equivalence classes. The investigation of geometric variational problems is thus of fundamental importance both in pure mathematics and in applications. This project is intended to significantly improve our knowledge on geometric variational problems by addressing a series of old and new questions, whose answer will require either the development of new techniques and ideas, or devising new approaches to known methods. This will be done through the collaborations with many leading experts in the field.This project intends to put a step forward in the study of geometric variational problems by dealing with the following lines of research. Inspired by work of L. Simon, the investigator proposes to study the optimal regularity of the singular set for minimal surfaces close to special classes of minimal cones and to construct new examples of singular area minimizing hypersurfaces. In the free-boundary setting, following ideas of Caffarelli and Weiss, the investigator proposes to study the regularity of the singular set of solutions to the Alt-Caffarelli functional and the thin-obstacle problem. Finally, in a joint project with Y. Liokumovich, the investigator proposes the problem of constructing smooth minimal hypersurfaces for a generic set of metrics on a compact manifold, generalizing works of Hardt-Simon and N. Smale. These problems are of fundamental nature, as they could lead to extensions of theorems such as Colding-Minicozzi's proof of the finite time extinction of the Ricci flow, Marques-Neves' proof of Willmore's conjecture, or Schoen-Yau's proof of the positive mass theorem.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

对几何变量问题的研究 - 这里的“问题”意味着找到一个将一些几何定义的能量概念最小化的对象 - 是数学中最古老，最迷人的主题之一，至少可以追溯到1932年在这一行列中授予J. Douglas的首个“诺贝尔奖”（诺贝尔奖学金）。几何变量问题的解决方案可以描述物理系统或社会经济模型的平衡配置 - 我们期望现实世界系统自然达到最小能源配置的情况。同样，在拓扑的数学领域（无论它们如何弯曲或伸展，对象都被视为等效），而变异问题的解决方案可以为如此大的等价类别提供优先的代表。因此，对几何变异问题的研究在纯数学和应用中都至关重要。该项目旨在通过解决一系列新旧问题来显着提高我们对几何变异问题的知识，这些问题的答案将需要开发新技术和思想，或者为已知方法设计新方法。这将通过与该领域的许多主要专家的合作来完成。该项目打算通过处理以下研究渠道来研究几何变异问题。受西蒙·西蒙（L. Simon）的作品的启发，研究人员建议研究奇异集的最佳规律性，以靠近特殊类别的最小锥体，并构建新的奇异区域的新例子，以最大程度地减少曲面。在自由边界的环境中，遵循咖啡雷利和魏斯的想法，研究人员建议研究alt-caffarelli功能和薄型问题的单一解决方案的规律性。最后，在与Y. Liokumovich的联合项目中，研究人员提出了一个问题，即在紧凑的歧管上为一组通用指标构建平滑最小的超曲面，从而推广Hardt-Simon和N. Smale的作品。这些问题具有基本的性质，因为它们可能导致定理的扩展，例如微微尼齐兹（Minicozzi）证明了Ricci流动的有限时间，Marques-neves的证据证明了Willmore的猜想，或者Schoen-yau的正式理论证明了NSF的构建机构。更广泛的影响审查标准。